Asymptotic expansion of the difference of two Mahler measures

2011/11/01 by Condon, John D.
#11C08 (Primary) 11R06 #41A60 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1111.0241

Abstract

We show that for almost every polynomial P(x,y) with complex coefficients, the difference of the logarithmic Mahler measures of P(x,y) and P(x,xⁿ) can be expanded in a type of formal series similar to an asymptotic power series expansion in powers of 1/n. This generalizes a result of Boyd. We also show that such an expansion is unique and provide a formula for its coefficients. When P has algebraic coefficients, the coefficients in the expansion are linear combinations of polylogarithms of algebraic numbers, with algebraic coefficients.

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